MODIFIED CONVEX HULL PRICING FOR FIXED LOAD POWER MARKETS. Vadim Borokhov 1. En+ Development, Shepkina 3, Moscow, , Russia

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1 MODIFIED CONVEX HULL PRICING FOR FIXED LOAD POWER MARKETS Vadm Boroov En+ Development Sepna Mosow 99 Russa Abstrat We onsder fxed load power maret wt non-onvextes orgnatng from start-up and noload osts of generators Te onvex ull (mnmal uplft) prng metod results n power pres mnmzng te total uplft payments to generators w ompensate ter potental profts lost by aeptng entralzed dspat soluton treatng as foregone all opportuntes to supply any oter output volume allowed by generator nternal onstrants For ea generator we defne a set of output volumes w are eonomally and tenologally feasble n te absene of entralzed dspat and propose to exlude output volumes outsde te set from lost proft alulatons New prng metod results n generally dfferent set of maret pres and lower (or equal) total uplft payment ompared to onvex ull prng algortm Keywords: power prng onvex ull prng mnmal uplft prng prng non-onvextes I Introduton Lberalzaton of power setors paved te way to development of eletrty marets wt free prng for power w an be eter entrally oordnated or deentralzed eg based on blateral trade Te entrally oordnated eletrty marets are often based on seurty-onstraned ommtment and eonom dspat optmzaton models w are redued to te least ost ommtment and dspat models for power systems wt fxed load e perfetly nelast demand If te entralzed dspat optmzaton problem s onvex ten tere exsts equlbrum pre (not neessarly unque) tat supports te soluton: gven te pre no maret player atng as pre-taer as eonom nentves to dstort ts output/onsumpton volumes Te equlbrum pre f t exsts an be obtaned by means of Walrasan auton were ea maret player submts ts supply/demand volumes at every possble pre and te maret pre s ten set so tat te total demand equals te total supply Gven te onsumer/produer beneft/ost funtons te total supply/demand volumes at a gven pre s determned usng deentralzed dspat problem obtaned by Lagrangan relaxaton of power balane onstrant n entralzed dspat problem For gven fxed unt ommtment statuses margnal prng metod for onvex entralzed dspat problems [] [] provdes equlbrum pre (set of pres) w ensure (sort-term) stablty of te maret outome e no generator/onsumer as eonom nentves to ange ts output/onsumpton volumes set by te entralzed dspat In unnode one-perod power maret wt onvex entralzed dspat problem tat pre (set of pres) s gven by nterseton of aggregate supply and aggregate demand urves However te margnal pre doesn t reflet non-onvex E-mal: vadmbor@yaooom Te vews expressed n ts paper are solely tose of te autor and not neessarly tose of En+ Development

2 features of power output su as non-zero mnmal apaty lmts fxed osts (su as start-up and no-load osts) as well as oter soures of non-onvextes and may not ompensate full ost of power output so tat oter prng meansms are needed to ensure bot eonom stablty of entralzed dspat outome and nononfsatory maret prng for power (for example sde-payments to generators) If unt ommtment problem s norporated n te entralzed dspat problem ten due to ntegral nature of ommtment desons and abovementoned non-onvex aspets of power output te margnal pre may fal to be equlbrum maret pre moreover equlbrum maret pre may not even exst (for example due to absene of a pre w supports maret outome) and oter meansms are needed to ensure stablty of entralzed dspat soluton Dfferent prng semes ad been proposed for marets wt non-onvextes [4]-[8] nludng ntroduton of new produts/serves (and assoated pres) n addton to eletr power utlzaton of nonlnear prng metodology (wt generator revenues beng nonlnear funtons of power output) applaton of unform (lnear) prng for power wt applable uplfts (sde-payments) Convex ull prng developed n [4] [5] and [9] stays wtn lnear prng framewor wt uplft payments (w are generally nonlnear funtons of output/onsumpton) ntrodued to ensure stablty of entralzed dspat soluton It s assumed tat ea maret player as an opportunty to supply/onsume any power volume satsfyng ts nternal onstrants In ts framewor ea maret player s ompensated te proft lost due to followng te entralzed dspat soluton Maret player lost proft s alulated as te dfferene between proft nferred from ndvdual maret player deentralzed dspat soluton and ts proft n entralzed dspat solutons at a gven maret pre Te total lost proft (and ene te total uplft needed to stablze te entralzed dspat) equals te dualty gap emergng after Lagrange relaxaton proedure s appled to power balane onstrant [4] [5] [9] Te onvex ull prng metod produes pres w mnmze total uplft payment needed to ompensate maret players for tese foregone opportuntes Te uplft alloaton may results n onfsaton on supply/demand sdes and/or dstorton of maret player bds Te latter may tae plae f uplft arges are alloated among produers/onsumers n a way redung/nflatng ter revenues/ expenses but preventng onfsaton In tat ase produers ave eonom nentves to nflate power output osts n ter bds wle onsumers are motvated to ndate redued beneft from power onsumpton (provded tat output/onsumpton volumes leared by te maret are unanged) to sow lose to zero proft obtaned at te maret and avod uplft arge alloaton Also large uplft payments may result n maret power abuse by maret players Moreover uplfts derease transpareny of te maret prng and suppress eonom sgnals Terefore t s all-mportant to redue total uplft payment needed to support te entralzed dspat soluton An approa to aeve tat based on ntroduton of addtonal redundant onstrant and assoated pre was proposed n [9] Te resultng uplft s redued ompared to [9] at te expense of avng new serve (a unt beng n a state ON ) and assoated pre ntrodued at te maret Te redundant nature of extended onstrant set ndates tat te feasble sets of te prmal problems spefed by orgnal onstrant set and extended onstrant set are dental Te need to ntrodue addtonal serve and assoated pre stems from te fat tat te newly ntrodued onstrant depends on optmzaton varables relatng to more tan one generator w leads to a need to ntrodue te assoated pre n te Lagrange relaxaton proedure Also sne te new onstrant as te form of nequalty te total uplft

3 payment s not equal te dualty ga ene mnmzng total uplft payment s not equvalent to solvng te orrespondng dual problem In te present paper we onsder te total uplft reduton problem wtn lnear prng framewor wt eletr power beng te only traded ommodty e no extra produts/serves and assoated pres are ntrodued Ea of te addtonally ntrodued redundant onstrants depends on an output of one generator only ene t ould be treated as nternal generator onstrant wtout a need to ntrodue te assoated Lagrange multpler n te dual problem formulaton Our proposal based on analyses of maret player opportuntes foregone by aeptng te entralzed dspat soluton exludes some uplft payments to maret players w we regard as exessve and tus redues (or leaves unanged) total uplft needed to support te entralzed dspat soluton ompared to onvex ull prng algortm developed n [4] [5] [9] For smplty we onsder one-perod power maret based on unnode power system e power system wtout transt losses networ and ntertemporal (su as rampng) onstrants We also assume zero mnmal apaty lmts of generators Te paper s organzed as follows We start wt a sort revew of onvex ull prng metod n Seton II and defne a set of opportuntes avalable to generators n deentralzed power maret n Seton III In Seton IV we formulate our proposal n terms of modfed onvex ull prng metod sow tat n onvex ase t s equvalent to standard margnal prng and dentfy lass of power systems for w onvex ull prng algortm and proposed metod may result n dfferent sets of pres Struture of te set of pres produed by modfed onvex ull prng metod s analyzed n Seton V Seton VI ontans examples of power systems wt omparsons of pres and assoated total uplft payments resultng from onvex ull prng and proposed metod Conlusons are presented n Seton VII II Convex ull prng Consder a entrally dspated one-perod unnode power maret wt fxed C X X u x ) I ( demand d and n generatng unts bddng ost funtons ) ( I { n} n wt output volumes x and bnary unt state varables u ea tang values n te set Z { } (wt for unt n a state OFF and for unt n a state ON ) Generator ost funtons are assumed to ave a struture C ( X) ( x ) wu wt fxed ost w w nondereasng onvex ontnuous funton ( x ) defned for x x x R were x denotes generator s mal apaty lmt ( x ) we also ave ( ) Te fxed ost orresponds eter to start-up and/or no-load ost for smplty we wll refer to t as start-up ost Intally all te unts are assumed to be n a state OFF Te entralzed dspat optmzaton problem (w we refer to as te prmal problem) wt optmzaton (deson) varables X X X ) taes te form X X G I x d I ( n mn C ( X ) () I

4 were s te total ost to meet demand and ea G denotes a set spefed by nternal onstrants of generator : G { X u Z x R x u x } Internal generator onstrants spefyng sets G I and power balane onstrant n () yeld te feasble set of te prmal problem denoted as w s assumed to be nonempty and ompat Let X ( X ) X ( u x ) I denote a soluton X n to () To proeed furter we state some nown matematal fats about optmzaton problems n queston Gven a maret pre p deentralzed dspat problem for generatng unt s formulated as ( p) [ px C ( X )] () X XG and defnes ndvdual supply urve of te unt We note tat sne ( p) s pont-wse mum of a funton lnear n p t s onvex n p wt well-defned subdfferental ( p) Sne dom ( p) R w s an open set ( p) s also emn ontnuous on R Let s defne x - mnmal eonom output as follows: f emn emn w ten x ; f w ten x s te lowest soluton (f any) to e mn emn emn emn equaton [ w ( x )]/ x ( x ) for x x ; f tere s no emn x x emn emn soluton ten x x (We note tat for x an be equvalently defned as te lowest soluton to any of e mn emn emn emn emn [ w ( x )]/ x ( x ) and {[ w ( x )]/ x } were emn denotes rgt dervatve) Tus x depends on generator s ost funton and ndvdual feasble set G only For any gven pre p te set of mzers of () as emn output volumes from te set {} [ x x ] Tus f start-up ost s nonzero ten te supply urve as a ga as te urve doesn t ave ponts wt output n te range emn ( x ) Te output volumes from tat range wll never be suppled n deentralzed dspat problem under any p (We note owever tat ( x emn ) ( p) for p [ w ( x emn start-up ost and lnear x ) we ave ( )]/ x emn emn ) For example n ase of nonzero x x and output mzng () equals eter zero and/or x dependng on te value of p tus output volumes from te open nterval ( x ) do not mze () at any maret pre However for [ w ( x )]/ x p we ave ( p) [ ] x Optmzaton over bnary varable u for gven value of x allows to exlude te bnary varable from te problem () at te expense of avng dsontnutes ntrodued n te ost funton: ( p) px f ( x ) () x x [; x ] We note te followng property of te gven formulaton of generator nternal onstrants: zero output s possble at bot unt ON and OFF states If start-up ost s nonzero ten state ON at zero output wll not be problem () outome If te start-up ost vanses ten outome of () wt unt zero output an ave any of two possble unt states w ndates redundany of state varable as optmzaton varable for te unt wt zero start-up ost 4

5 wt f x ) w ( x ) ( x ) dom f ( x ) [ x ] and step-funton x ) defned ( equal to for x and oterwse Let s defne f x ) outsde dom f x ) and extend feasble set n () to R ten ( p) s Fenel onvex onjugate of f x ) : ( An mportant property of Fenel onjugaton [] s tat f [ l( onv f ene ) ( x f )] ( f an be replaed by l onv f ( x )) - te greatest ( losed onvex funton majorzed by f x ) nown as losed onvex ull of f x ) ( w we denote by f x ) We ave dom f ( x ) [ x ] te funton f x ) s ( ontnuous on ( x ) taes nfnte values on ( ) ( x ) and s lower sem-ontnuous on R We note tat n general ase tat replaement results n a dfferent set of mzers for () Te funton f x ) an be formally obtaned by double Fenel onjugaton of f x ) [] Sne ( p) s proper ontnuous onvex ( funton on R applaton of Fenel Moreau teorem yelds ( p) ( p) ene ( f ( x )) Sne () stays vald f ( x ) ( x ) x ( p) p f ( x ) (4) ( f ( x represented as ( ( p) p) ( ( ( ( ( f and feasble set [ x ] are replaed by f and R respetvely nverson rule for subgradent relatons [] yelds Tus te set of output-pre ponts of te form x )) an be equvalently Tat set of ponts would be te supply urve for te unt f ts ost funton were f x ) Eq (4) mples tat we ave ( emn emn p [ w ( x )]/ x emn emn emn ( p) [ x ] p [ w ( x )]/ x (5) emn emn p [ w ( x )]/ x mn were x e s mal output volume x satsfyng [ w ( x )]/ x ( x ) for x x and te dots denote elements (w may depend on p ) no lower tan emn x We note tat f x ten (5) s well-defned beause bot ( ) and emn x as well as onvexty of ( x ) mply lm ( x )/ x () x w s fnte Now we are ready to formulate dual of te prmal optmzaton problem (): D mn pd x C ( X ) pd ( p) (6) pr X pr I I I X G I As RHS of (6) s unonstraned mzaton problem of onave funton ts set of mzers w we denote as P s gven by solutons to d ( p) It s stragtforward to see tat P s nonempty and ontans nonnegatve elements only Sne all ( p) are ontnuous funtons te objetve funton n mzaton problem (6) s also ontnuous It follows from () tat I and p su tat p ([ w ( x )]/ x ; ( x )) proft funton as a struture ( p) px O( p) were O ( p) denotes terms ndependent from p Tus for p ([ w ( x )]/ x ; ( x )) te objetve funton n (6) as te form I I 5

6 pd d I x x I O( p) and sne feasblty of te prmal problem () entals te followng two ases are possble If d ten objetve x I funton n (6) attans ts mal value for any p from te ray [ ([ w ( x I )]/ x elements are nonnegatve If ; ( x d x I )) ) ene P s nonempty and all of ts ten objetve funton n (6) s negatve for p zero at p and negatve for p ger tan ertan suffently large value ene te set of p wt nonnegatve values of objetve funton s bounded and ontans only nonnegatve values of p Tat set s also losed as te nverse mage of losed set [ ) under te ontnuous funton Tus n ts ase te feasble set of (6) an be redued to a subset wt nonnegatve values of te objetve funton w s a ompat set Applaton of te extreme value teorem ensures tat P s nonempty n tat ase as well Clearly n ts ase all elements of P are nonnegatve Let X ( u x ) denotes a mzer of te problem () wt p P Te dualty gap s gven by D [ ( p ) ( X )] p P (7) I and aordng to te approa developed n [4] [5] [9] represents sum of generator lost profts assoated wt opportuntes to supply power n volumes orrespondng to X X ) at a pre ( X n p P foregone by aeptng dspat X Tus dualty gap equals te total uplft and mnmzng te total uplft s dental to solvng dual problem (6) Te relaton between te total uplft and dualty gap stems from te fat te relaxed onstrant (e power balane onstrant) as te form of equalty Sne ( p) ( X ) I p P te dualty gap s nonnegatve If eter of () () and (6) ave multple optmal ponts te dualty gap s ndependent from te oe of X X p P Te onvex ull prng metod nstruts to dstrbute te amount (7) to generators as uplft payments to ensure tat no generator atng as a pre-taer (e leavng asde ssues related to exerse of maret power) as an nentve to ange ts output ( u ) gven te maret pre p P Te followng two nterpretatons are x applable to te set P On one and (7) entals tat te set of pres s osen n a way to mnmze te total uplft payment needed to support te entralzed dspat soluton [4] On te oter and te set P an be vewed as subdfferental of te D onvex funton w s onvex ull of total ost funton vewed as funton of d [5] [9] Tat justfes te terms mnmal uplft prng and onvex ull prng used to desrbe te metod An mportant attratve property of te pres resultng from onvex ull prng s tat tey are monotonally nreasng n load sne subdfferental of onvex funton s a non-dereasng operator; owever due to uplft payments te aggregate generator revenue s generally not a monotonous funton of load [5] A pre p s sad to support te soluton X f 6

7 X arg X XG px C( X ) I It s stragtforward to verfy tat a pre tat supports soluton to te prmal problem exsts ff te dualty gap s zero In tat ase te set of pres supportng soluton X s te same for all X (f te prmal problem as multple solutons) and s dental to a set of mzers for dual optmzaton problem (6) Te aggregate supply at a gven pre s a sum of generator outputs obtaned by solvng te orrespondng deentralzed dspat problems at tat pre Te dual problem provdes a framewor to fnd a maret pre (set of pres) w orresponds to a transton from sortage of aggregate supply ompared to demand d to surplus of aggregate supply over d If aggregate supply and demand urves nterset at some p ten power balane onstrant olds at te nterseton segment and X (some X f () as multple solutons) belongs to feasble set of te prmal problem ene te dualty gap s zero (Here we use te fat tat power balane onstrant relaxed n dual problem formulaton as te form of equalty If nequalty onstrant s relaxed and some optmal pont of relaxed problem s feasble n prmal problem ten dualty gap ould stll be present) Te onverse s also true: f dualty gap s zero ten p supports X and aggregate supply and demand urves nterset at te pre p beause X satsfes power balane onstrant Terefore dualty gap n te model under onsderaton ours only f te aggregate supply and demand urves don t nterset Tus n ase of unnode one-perod power system under onsderaton te onvex ull prng produes a maret pre (a set of pres) w orresponds to eter nterseton of aggregate supply and demand urves (no dualty gap) or transton from undersupply (demand exeeds aggregate supply volume) to oversupply (aggregate supply volume exeeds demand) Note tat () provdes a stragtforward way to fnd set of pres for oneperod unnode power system resultng from te onvex ull prng metod: replae generator ost funtons C ( X ) by f ( x ) n te prmal problem tat results n onvex entralzed dspat problem and fnd te orrespondng set of margnal pres Grapally P orresponds to nterseton of demand urve wt aggregate supply urve of te new problem onstruted from ndvdual supply urves ( f ( x ); x ) of generators wt onvex ost funtons f ( x ) To llustrate some mplatons of te onvex ull prng we onsder te followng example Compreensve study of mportant propertes of onvex ull prng metod was presented n [] Example Consder power system wt fxed demand d and a sngle suppler avng ost funton C( X ) wu ax wt start-up ost w and onstant margnal ost a zero mnmal output lmt and mal output lmt x w s assumed to exeed demand Clearly te prmal problem soluton yelds u x d wt margnal pre a Supply urve reonstruted from generator s deentralzed dspat problem s omprsed of two dsjont segments x for p a w/ x and x x for p a w/ x and s terefore dsontnuous We also observe tat supply urve doesn t ave a pont wt output equal demand and ene supply and demand urves 7

8 don t nterset Also generator s output x s below emn x w equals example Applaton of onvex ull prng metod yelds sngleton set p a w/ x x n ts P wt a pre p w s below generator s average ost for output d Te pre mples two possbltes for te unt state-output varables: u x and u x x bot yeldng ( p X ) Hene te generator s ompensated wt uplft payment of w( d / x ) w results n zero generator proft for output d We note tat f te pre were set above p ten aordng to te onvex ull prng prnple generator would ave to be ompensated for te foregone opportunty to proftably supply x w nreases uplft payment On te oter and generator s not able to supply any non-zero volume oter tan d and generator ad not lost any opportunty to supply any ger output by aeptng te entralzed dspat soluton Tus x -dependene of te pre seems ounterntutve Also sne te pre p s below generator average output ost t may deter new potental supply enterng te maret able to fully replae te numbent generator beause underestmates te level of average output ost needed to suessfully ompete wt te produer It seems more desrable to ave pre ndependent from nfeasble output volumes e volumes above d Tat s aeved for example wen pre s set to ( a w/ d) n tat ase no uplft payment s needed at all f nfeasble output volumes are removed from lost proft alulaton Te reason for x -dependene of emn p s tat d x x and possble way to ave loser to d optmal output n dual problem s to redue emn x by lowerng x Te ompensaton of te lost proft due to foregone opportuntes mples tat te maret player ould reeve tat addtonal proft f not for te entralzed dspat Te example above llustrates tat some opportuntes treated as foregone n onvex ull prng metod annot be realzed by maret players n deentralzed maret and ene opportuntes avalable to generators sould be examned n more detals III Opportuntes avalable to maret players Lost proft ompensatons assoated wt foregone opportuntes are needed to ensure stablty of entralzed dspat outome Tat mples tat for ea generator tere s number of avalable legtmate atons t may undertae to dstort outome of () n order to reover ts lost proft Tus to alulate te requred ompensaton one needs to determne te set of generator output volumes resultng from tose atons Let s allow generator to engage n blateral ontrats for power wt te oter maret partpants (bot generators and onsumers) payng to oter generators full ost of ontrated output volumes aordng to ter bds and reevng payments from onsumers for te ontrated volumes n te amount ndated n ter bds Te maret player bds used n te desrbed proedure are te ones submtted for te entrally oordnated maret Sne demand s fxed we formally requre all demand volumes to be fully ontrated tus te feasble set of te optmzaton problem s unanged We requre te resultng output/onsumpton sedule to be bot feasble p 8

9 and attanable as entralzed dspat outome In ts settng only maret partpants wt ontrats are allowed to partpate n entralzed dspat optmzaton problem usng ter orgnal bds wt volumes restrted by te ontrated output/onsumpton submts ts bd for a volume equal te netted ontrated volumes (te generator volume e ts ontrated output) Te total fnanal effet of generator from all te ontrats sould be nonnegatve Formally we say tat generator as an opportunty to supply output volume x x x f tere exst a set of x I { } and a set of x x \ orrespondng u I so tat X ( X X n ) wt X ( u x ) satsfes te followng ondton: X arg mn C ( X ) (8) X X X X Tus te feasble set n (8) s gven by tat of () supplemented by I u u x x I If x ten u ; f x ten u ; f bot w and x ten we ave eter u or u Clearly we ave X Let s denote by all X satsfyng (8) Sne te generator asde from produng output ats as ntermedary (retanng te maret surplus) te set oe of To mae a transton from gven produer let s denote as olleton of of but X on te set I с I te set of x merely s ndependent from te to a set of possble output volumes for a с a set ontanng all so tat X Te set Z R orrespondng to X In general ase X su tat tere exst a с an be vewed as projeton G с We also note tat te prmal problem outome an be realzed troug a set of blateral ontrats and terefore с X X Sne all X are prmal feasble te rgt-and sde of (8) defnes selforrespondene wt a range It follows from (8) tat s a set of fxed ponts of tat orrespondene Let s defne as N all elements X from wt x u at least for one su tat w (f tere are no su elements ten N s an empty set) Denote as te set exludng N ten all elements of an be realzed troug a set of blateral ontrats: Proposton : Proof Clearly we ave sne te orrespondng x satsfes so we need to sow tat I x d and x x с I с Let X I we onlude tat X satsfes (8) for X beause X X s te only feasble pont n optmzaton problem (8) Tus we ave and ene Proposton s proved Proposton allows to fnd expltly by projetng on assoated wt X : te orrespondng values of output volumes с Z R x are gven by 9

10 mn mn losed nterval [ x x ] wt x ( d x ;) : I x mn ( d; x ) Tus asde from speal ases wen eter generator zero output and/or mal generator output are nfeasble te set of possble output volumes by te generator s dental to tat spefed by G (up to a ponts ( u x ) for w ) and modfed onvex ull prng presented below s dental to onvex ull prng approa We note tat f N s nonempty ten and ene are not ompat Te set с mn s nonompat ff bot w and x In tat ase we ave с {()} {( u x ) u x x } Tat rases a queston f mum с (mnmum) of a ontnuous funton of X exsts on Sne с {( )} s ompat ten te funton ontnuous on с {( )} s guaranteed to ave extremum on t and f te extremum s attanable outsde te pont u x ) ( ten te answer s postve Wen start-up osts of all unts vans optmzaton of te bnary state varables n prmal problem produes onvex problem In ts ase beomes a set of possble output volumes IV Modfed onvex ull prng Let s defne modfed prmal problem mn x and equals te losed nterval [ x x ] ( ) mn C ( X ) (9) X XG ( ) I x d I wt G ( ) {()} ( X ) were X I X ) {( u x ) ( u x ) G x x } () ( wt some > ( n ) Inluson of ( X ) for ea element X s needed to ndate n te dual problem weter at a gven pre generator s wllng to supply some more/less power tan x ompatble wt ts nternal onstrants Tus for ea x wt X sets G ( ) and G ave dental output volumes n some losed negborood of x for ponts n tat negborood ompatble wt G Regardng te need to ensure tat G ( ) nludes a pont ( u x ) we ave te followng omment If nludes only elements wt unt avng output no lower tan some postve value ten for suffently small all elements of ( X ) X orrespond to unt s state ON and f ( u x ) s not nluded n G ( ) ten n deentralzed dspat problem for a gven pre te unt wll fnd ts optmal output volume dsregardng te start-up ost w Hene n ts ase w ontrbutes neter to с

11 P ( ) nor uplft payment for te unt w may result n negatve proft for te generator mplyng onfsatory prng By onstruton we ave G ( ) G Let s denote by ) ( te feasble set of te modfed prmal problem (9) Sne X ( ) and ( ) for any X - mzer of te prmal problem () we onlude tat ( ) and bot prmal and modfed prmal problems ave dental sets of mzers Sne we ave explt expresson for t s also possble to formulate G ( ) expltly : f mn mn x ten for suffently small su tat : x mn ( ) {()} {( ) mn[ ; u x u x x x x mn mn x and x or f x mn ): ( ) {()} {( ) mn[ ; u x u x x x ]} G ]}; oterwse (e f bot G Tese expressons also llustrate a need to ntrodue Consder deentralzed dspat problem for unt wt feasble set G ( ) for some fxed maret pre If and x mn s optmal unt output ten t means tat generator eter sells all output volumes at a pre no lower tan margnal ost of output or generator maes nonnegatve proft but sells some part of output volumes below ts margnal ost and would prefer to derease ts output If and x s optmal unt output ten t ould mean tat eter generator operates at mal apaty (e x x ) and wll not ange ts output f te maret pre nreases or te generator wll nrease ts output f te pre nreases Introduton of > allows to dfferentate between tese ases We also note tat te set G ( ) s ompat I Dual of te modfed prmal problem as te form D ( ) pd ( ) pr wt ( ) ( X ) () X I X G ( ) We note tat aordng to () te set G ( ) for a unt wt w may nlude eonomally nfeasble pont ( u x ) If () s modfed to exlude tat pont from ( X ) for su a unt te resultng set G ( ) beomes nonompat However sne for unt wt w te pont ( u x ) neter belongs to mnmzer of (modfed) prmal problem nor mzes ( ) nluson of tat pont n ( X ) affets neter of ( ) D ( ) and ( ) Tus we onlude tat su modfaton of ( X ) doesn t ange te set of maret pres obtaned from () or ndvdual generator uplfts Lewse te prng outomes are not affeted f te pont ( u x ) n defnton of G ( ) s substtuted or supplemented by ( u x ) for generator wt w Sne ea ( ) s pont-wse mum of te funton lnear n p t s onvex n p wt well-defned subdfferentals wt respet to p w we denote as ( ) Let P ( ) be a set of mzers of () It s stragtforward to verfy

12 tat P ( ) s nonempty Clearly P ( ) s a set of pres w solve d ( ) Relaton G ( ) G mples D D ( ) Hene I D D ( ) ( ) w entals relaton between dualty gaps of orgnal and modfed optmzaton problems: D D ( ) ( ) Terefore total uplft needed to support entralzed dspat soluton at any pre p P ( ) n modfed optmzaton problem (9) s no ger tan tat for orgnal problem () at any pre p P Moreover f dualty gap of te orgnal problem s zero ten dualty gap of te modfed problem s also zero (Te onverse s generally not true as t s llustrated n Example below) We propose to alulate te set P ( ) and lost profts ( ) ( p p X ) for p P ( ) n te lmt as and utlze tem as te set of maret pres and ndvdual generator uplfts respetvely Clearly te ndvdual uplfts are ndependent from te oe of p P ( ) Let's onsder te ase of no generator start-up osts ( w I ) Te dualty gap s zero as te prmal optmzaton problem beomes onvex after exluson of te bnary state varables We prove tat n ts ase te set of pres obtaned from onvex ull prng metod s dental to te set of pres gven by modfed onvex ull prng algortm Proposton : Let for some we ave bot w and ten a pre p supports X n deentralzed dspat problem ff t supports soluton deentralzed dspat problem: X Proof Clearly sne arg X XG X px C( X ) X arg px C( X ) G ( ) G f p supports X X G ( ) X n modfed X n deentralzed dspat problem ten t supports X n modfed deentralzed dspat problem To prove te onverse we note tat te bnary state varables an be exluded from bot deentralzed dspat problem and modfed deentralzed dspat problem for te unt - we denote te resultng feasble sets as g and g ( ) respetvely Clearly we ave g { x x R x x } and te generator deentralzed dspat problem beomes onvex Let p supports x n te modfed deentralzed dspat problem () for generator ten sets g ( ) and g are dental n te losed - negborood of x (te -negborood may belong neter to g ( ) nor to g but bot g ( ) and g ave nonempty nterseton wt te -negborood If x or x x ten x belongs to te boundary of tat nterseton) Hene te onave funton px ( x ) as loal mum at x on a onvex set g Terefore t as global mum at x on g w entals tat p supports x n te deentralzed dspat problem () Proposton s proved Hene f start-up ost of all generators vans ten te onvex ull prng and modfed onvex ull prng metods result n dental sets of pres e margnal pres We note tat te modfed onvex prng approa n ts ase

13 produes a set of maret pres w s ndependent from We also note tat с Proposton stays vald f X s replaed by X mn Now we return to non-onvex ase and observe tat x an be formally set to zero n dual of te modfed prmal problem e extendng G ( ) to nlude all te elements of G wt output volumes n te range [ x ] affets neter te set of maret pres nor te uplft reeved by ea generatng unt n modfed onvex ull prng framewor Defne D ˆ ( ) pd ˆ ( ) () pr I wt ˆ ( ) ( X ) ˆ G ( ) {( u x ) ( u x ) G x x } X X Gˆ ( ) Let s denote by ˆ P ( ) te set of mzers of () Proposton 4: For optmzaton problems () and () wt I we ave ˆ P ( ) P ( ) ; ˆ ( ) ( ) p P ( ) I ; D D ˆ ( ) ( ) Proof At frst we study relaton between ( ) and ˆ ( ) If for gven we mn ave x ten G ( ) ˆ G ( ) and ene ˆ ( ) ( ) p R terefore ( ) an be replaed by ˆ ( ) n () wt no effet on P ( ) or D ( ) mn Oterwse e f x let s defne f ( x ) as f( x ) f( x ) for x orrespondng to output volumes n G ( ) and oterwse lewse defne f ˆ ( x ) usng output volumes n ˆ mn G ( ) For x and x x we ave f ( ) ˆ x f( x ) Consder ter respetve onvex ulls f ( x ) and ˆ f ( x ) w are onvex funtons avng fnte values on x mn( x ; x ) It s stragtforward to verfy tat ( ) ˆ mn f x f ( x ) for x and x x Tus ( ) ˆ f x f ( x ) for x mn x Sne ea ˆ ( ) s pont-wse mum of te funton lnear n p t s onvex n p wt well-defned ˆ ( ) - subdfferentals wt respet to p Usng ˆ ( ) ˆ ˆ ( ) { x x dom f x p f ( x )} () and analogous expresson for ( ) we onlude tat n te range mn x x mn( x ; x ) sets ( ) and ˆ ( ) ave dental elements (f any) p R As () s mzaton problem of onave funton ts set of mzers ˆ P ( ) s gven by d ˆ ( ) and ene elements of ˆ ( ) lower tan I x (f any) don t affet te set ˆ P ( ) analogous observaton olds for P ( ) mn

14 Tus wen I solutons to d ( ) and d ˆ ( ) are dental terefore ˆ P ( ) P ( ) mn If x ten te seond bullet s trvally satsfed Tus we fous on te mn mn ase x Te defnton of x mples tat f ( ) as no elements equal mn or above x ten d ( ) and p P ( ) Hene p P ( ) bot sets I ( mn ) and ˆ ( ) restrted to [ x mn( x ; x )] are nonempty As we ave seen above tese sets are equal wen lmted to mn ( x mn( x ; x )] wt Terefore p P ( ) restrtons of ( mn ) and ˆ ( ) to [ x mn( x ; x )] are nonempty and equal As px f ( x ) and ˆ px f ( x ) are onave funtons on onvex set x mn( x ; x ) sets ( ) and ˆ ( ) are mzers of ( ) and ˆ ( ) respetvely Sne ( ) ˆ f x f ( x ) for x mn x we readly obtan ˆ ( ) ( ) ( mn p P ) for te ase x Statements of te frst two bullets trvally mply lam of te trd bullet Proposton s proved mn Anoter vew on te Proposton 4 s te followng If x ten G ( ) Gˆ ( ) ( mn emn ) ˆ ( ) p R If x mn( x ; x ) emn ten sne output volumes from te open nterval (mn( x ; x )) never mze eter ( ) or ˆ ( ) we onlude tat ˆ ( ) ( ) p R At mn emn last f x x ten Proposton 4 mples tat values of p for w ( ) and ˆ ( ) mgt be dfferent do not belong to ˆ P ( ) P ( ) mn Havng sowed tat x an be exluded from onsderaton n dual of te modfed prmal problem we turn to ases wen x an be dsregarded as well e mn Defne I { I x x } ˆ e mn I { I x x } I D ( ) pd ( p) ˆ ( ) (4) pr I I ˆ and let P ( ) denote a set of mzers of (4) Proposton 5: For optmzaton problems () and (4) for I we ave ˆ P ( ) P ( ) ; p) ˆ ( ) ˆ p P ( ) I ; ( D D ( ) ˆ ( ) Proof Clearly f for a gven I we ave x x ten G ˆ ( ) G ( ) and ˆ ( p) ( ) p R and ˆ ( ) an be replaed by ( p) n () wt no effet on ˆ P ( ) or ˆ D ( ) Tus we may restrt our onsderaton to te ase of x x I It s stragtforward to verfy tat n tat ase ˆ f ( x ) f ( x ) for x ( ] and ene ˆ f x ) f ( x ) for x ( ; ) Usng x ( x I 4

15 (4) and () for I we onlude tat n te nterval ( x ) and ene n te range [ x ) sets ( p) and ˆ ( ) ave dental elements p R Sne (4) s mzaton problem of onave objetve funton P ( ) s gven by a set of solutons to d I ( p) ˆ ( ) and ene elements of ( p) ger tan x (f any) don t affet te set P ( ) Terefore for I equatons d ( p) ˆ ( ) and d ˆ ( ) ave dental I Iˆ Iˆ sets of soluton for p w entals ˆ P ( ) P ( ) Sne elements of ˆ ( ) ger tan d ˆ ( ) (we stll onsder te ase x I I x do not ontrbute to x ) we onlude tat I ˆ p P ( ) te set ˆ ( ) as at least one element n te range x [ x ] ene ˆ p P ( ) bot ˆ ( ) and ( p) are nonempty and equal wen restrted to tat range As bot px f x ) and ˆ px f ( x ) are onave funtons on onvex set ( x x sets ( p) and ˆ ( ) are mzers of ( p) and ˆ ( ) respetvely Sne ˆ f ( x ) f ( x ) for x [ x ] we onlude tat ˆ ( ) ( p) I ˆ D D p P ( ) w also entals ( ) ˆ ( ) Proposton s proved Tus Propostons 4 and 5 mply tat for I we ave ˆ P ( ) P ( ) P ( ) Sne ea unt from te set Î as pysal apablty to satsfy demand alone and exbts natural monopoly beavor (as dereasng average ost funton) for output volumes n ( d ] we wll refer to tese unts as large natural monopoly generatng unts (LNMGUs) Tus n te absene of LNMGUs onvex ull prng [4] [5] [9] and proposed modfed onvex ull prng result n dental sets of maret pres Tese prng metods may produe dfferent sets of pres only f te e mn system as at least one LNMGU We note tat x x mples x e mn d w entals d Trougout te rest of te paper Iˆ we oose suffently small x so tat x e mn d Tat ensures dereasng average ost funton for LNMGU output n te range d ] ( Generally dual problem (6) s onvex ene P s onvex Sne P s nonempty and ontans nonnegatve elements only t as one of te followng forms: a sngleton {a} a bounded losed nterval of te form [ a b] a ray [ a ) wt some a and b : a b R a b However f at least one LNGMU s present n te system ten te last possblty s not realzed sne for any pre no lower tan average ost of output for ts mal apaty tere s exess of supply over demand Terefore a set P s bounded n te presene of at least one LNMGU and s eter a sngleton or a bounded losed nterval Ea LNMGU s able to satsfy demand as x d owever on one and f ts start-up ost s too g ten t mgt be eonomally ratonal not to operate te unt (provded tat su a dspat s feasble) on te oter and nequalty 5

16 e mn x x prevents te start-up ost from beng too low w lmts te number of su unts n operatng state Tus natural queston s ow many LNMGUs an operate smultaneously aordng to te prmal problem soluton? We sow tat te answer s tat at most one Proposton 6: Any soluton to prmal problem () as no more tan one LNMGU avng nonzero output e at most one unt wt x e mn d as u Proof Assume te ontrary: let tere be a soluton to prmal problem () wt more tan one operatng unt wt x e mn d denote all su unts as M Sne (x) s onvex we ave ( x ) [ ( x ) ( x )] for x x I (5) were denotes left dervatve Let be a unt from M wt gest rgt dervatve at d : ( d) ( d) M Denote by S a set of all unts wt u exludng unt We ave [ w S { } ( x )] Let s onsder te prmal problem supplemented by addtonal onstrants: u for S (w also mply u ) u for S and denote by and te new prmal problem (f tere are multple mzers x te soluton and optmal output volumes of tem) Sne by assumpton tere s at least one more unt wt x denotes any one of x d and u te new problem s feasble and Clearly we ave w entals w ( x ) w ( x ) Let s denote by S z and S m subsets of S S { } S { } S S ontanng all unts wt x and x denote by S b tus we ave S S S z x m b respetvely te rest of unts n S we S Terefore ( x ) ( x ) ( x ) ( x ) w ( x ) ( x ) (6) S m S All x ) are onvex funtons terefore ( ( x ) ( x ) ( x )( x x ) b S z ( x ) () x ( x (7) ( x ) ( x ) a ( x x ) a ) (8) Sne values of all bnary varables n new prmal problem are set by onstrants ter values an be substtuted n te objetve funton and te rest of onstrants Te resultng new optmzaton problem s onvex sne t as ontnuous optmzaton varables only onvex objetve funton and lnear onstrants Te latter ensures tat lnearty onstrant qualfaton olds ene Karus Kun Tuer ondtons mply exstene of su tat Inequalty ) ( x m x e mn d usng (6)-(9) and S ; () Sz ; ( x ) Sb (9) and defnton of x mply [ w ( d)]/ d ( d) Hene S x d we obtan emn of ( x ) ( d) ( d)( x d) entals d ( d) ( d) ( x ) x ( d) x x mples w and u yelds x Terefore M \{ } belong eter to S z or to b Applaton Condton x e mn d (d) All unts from S ene for M \{ } we ave eter 6

17 () or ( x ) Usng ( ) ( x ) ( d) and (5) we onlude tat (d) M \{ } Te oe of mples (d) w s nompatble wt Tus tere s no more tan one unt wt x e mn d (d) n a state ON aordng to soluton to te prmal problem () Proposton s proved Sne prmal problem () and modfed prmal problem (9) ave dental sets of solutons and ea LNMGU as x d statement of Proposton 6 s true for modfed prmal problem (9) as well Now we rase tat queston n te ontext of dual problem Let s defne Iˆ ( ) { Iˆ[ w ( d )]/( d ) mn w ( d ) /( d mn ˆ Iˆ )} Proposton 7: If I ˆ ten for gven suffently small postve values of { } Iˆ removal of any group of LNMGUs from onsderaton n te dual problem (4) provded tat at least one LNMGU from I ˆmn ( ) remans wll not ange te set of maret pres resultng from (4) Proof p P ( ) we ave d ( p) ˆ ( ) () I For suffently small Iˆ applaton of (4) yelds: I ˆ p [ w ( d )]/( d ) ˆ ( ) [ d ] p [ w ( d )]/( d ) () d p [ w ( d )]/( d ) Hene f for some Iˆ we ave p [ w ( d )]/( d ) ten ˆ ( ) d d and sne all elements of ( p) are nonnegatve () mples p P () Terefore p mn w ( d ) /( d ) p P ( ) and all ˆ Iˆ LNMGUs wt w ( d )]/( d ) [ ˆ ˆ ˆ above mn w ( d ) /( d ) ˆ Iˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ do not ontrbute to RHS of () and an be dsregarded n (4) wtout affetng P ( ) Also f tere s more tan one LNMGU wt w ( d )]/( d ) mn w ( d ) /( d ) [ ˆ ˆ ˆ ˆ ˆ ˆ I ten all but any one of su LNMGUs an be removed from onsderaton from (4) wtout affetng te set P ( ) Proposton s proved As a ollorary we onlude tat no LNMGU as postve proft n dual problem (4) also all LNMGUs exept for any one belongng to I ˆmn ( ) (for gven values of Iˆ ) an be removed from onsderaton n dual problem (4) wtout affetng te set of maret pres P ( ) and ndvdual uplfts of all generatng unts We note tat statement of Proposton 7 and te ollorary are also vald for dual problems (6) () () wt possbly dfferent relevant LNMGUs Tus f more tan one LNMGU s present n te power system ten to alulate a set of maret pres usng modfed onvex ull prng for gven values of { } Iˆ we may dentfy LNMGU wt te lowest average ost [ w ( d )]/( d ) (f tere s more tan one LNMGU w satsfes tat property oose any of tem) and exlude te rest of LNMGUs from onsderaton 7

18 n (4) Note tat te set of LNMGUs wt te lowest average ost may depend on values of { } Iˆ We also note tat f at least one LNMGU s present n te power system ten te pre obtaned from (4) e any element of P ( ) (provded tat I ) annot exeed mṋ w ( d ) /( d ) wle any element of I emn e mn P annot exeed mn w ( x ) / x Clearly te upper bound for P ( ) s Iˆ no lower tan tat for P Tat s ompatble wt te fat tat proposed modfed onvex ull prng tends to produe ger pres ompared to onvex ull prng w s formalzed n Proposton 8 below Terefore at most two LNMGUs are relevant: one for prmal problem soluton for power output and te oter for dual problem soluton for pres Also LNMGU n a state ON n soluton for prmal problem () w of ourse s also a soluton for te modfed prmal problem (9) and LNMGU settng te pre n (4) mgt be dfferent (even f soluton to te prmal problem s unque) as t s sown n te example below Example Consder one-perod unnode power system wt fxed demand d and tree generatng unts g wt ost funtons gven by: C ( X) x d / 4; C( X ) d u x / x d ; C( X) 4d u x d Tus g and g ave emn zero varable ost of output It s stragtforward to verfy tat x x e mn d x e mn d Terefore unts g and g are LNMGUs Te prmal problem () as well as te modfed prmal problem (9) ave unque optmal pont wt g avng output d / 4 g produng d / 4 and unt g beng n a state OFF Tus t s LNMGU g w s ON n te unque prmal problem soluton Soluton of te problem (4) wll mae g produe d / 4 and satsfy te rest of demand wt eter g or g dependng on w unt as te lowest average total ost for output volumes d and d respetvely: ( d ( d ) / ) /( d ) for g and 4d /( d ) for g For suffently low we ave ( d ( d ) / ) /( d ) 5d 5 O( ) w mples tat te average total ost of g s lower ene g s OFF and g sets te pre equal to ts average total ost of output for a supply volume d n soluton to (4) (sne g as zero proft at tat pre t an be eter n a state OFF or ON aordng to te soluton) Regardng outputs of g and g n soluton for dual problem (6) we note tat emn sne unt g as average total ost for output x equal to d w s above average total ost for output x e mn 7d of unt g te soluton of te dual problem (6) mples tat g s OFF wle g sets te maret pre and an be n eter state wt zero proft V Lmt of P ( ) as 8

19 Propostons 4 5 and 7 mply tat for I te set P ( ) s ndependent from { } I and ene may depend only on { } Iˆ Terefore f no LNMGU s present n te power system ten for I we ave P ( ) P and P ( ) s ndependent from Let s examne n more detal te struture of P ( ) for power system avng at least one LNGMU Defne redued aggregate supply urve as te aggregate supply urve of all te generatng unts exludng Î e omttng all LNMGUs Te followng tree ases are possble Frst ase s wen te redued aggregate supply urve as a pont wt output volume lower tan d at a pre mṋ w ( d) / d For any suffently small I Iˆ we ave [ w ( d )]/( d ) [ w ( d)] d ene any LNMGU from / I ˆmn ( ) beomes margnal n soluton for (4) and sets te pre Tus n ts ase P ( ) s a sngleton wt an element mn w ( d ) /( d ) w s gven by I ˆmn ( ) mnmum of fnte number of funtons ontnuous n I ˆmn ( ) and ene s ontnuous funton of As for all Iˆ te maret pre nreases and attans te value of mṋ w ( d) / d I Te seond ase s realzed wen at a pre mṋ w ( d) / d I mnmal supply volume on te redued aggregate supply urve equals d Here we ave two possbltes If for any pre below mṋ w ( d) / d all te output volumes on te I redued aggregate supply urve are below d ten analyss and onluson of te frst ase are applable Oterwse for suffently small values of Iˆ te set P ( ) s gven by bounded losed nterval [ a b( )] wt a beng nonnegatve real number ndependent from and b( ) mn w ( d ) /( d ) w s a I ˆmn ( ) ontnuous funton of As all P ( ) tends to [ a b()] wt b() mn w ( d) / d Iˆ Now we turn to te trd ase wen at a pre mṋ w ( d) / d I all te ponts on te redued aggregate supply urve ave output volume ger tan d Tat means tat for suffently small postve Iˆ all LNMGUs are rrelevant n dual problem (4) soluton for pres and terefore P ( ) P Relaton between set of maret pres resultng from onvex ull prng metod and proposed modfed approa s gven below Proposton 8: For p P ( ) te followng olds for suffently small I : eter p P or all elements of P are below p Proof From () we dedue ( p ) ˆ ( ) d ( p) ˆ ( ) p P ( ) I Iˆ I wt all left/rgt dervatves beng nonnegatve For suffently small Iˆ () entals tat f ˆ ( ) ten ˆ ( ) d ene ˆ ( ) Iˆ 9

20 p P ( ) For suffently small q R Iˆ usng (5) and () we arrve at bot ˆ ( q ) ( q) and ˆ ( q ) ( q) Terefore ( p ) d ( p) ( p) p P ( ) I I If for a gven p P ( ) we also ave ( p ) ( p) d ten owever Sne I I I ( p ( ) p) > Iˆ Iˆ I d ten p p d ( p ( ( ) ( ) ) p) p P () Iˆ I Iˆ Iˆ Iˆ p P If ( p) ( p) s a onvex funton and subdfferental of a onvex funton s a monotone operator () entals VI Examples p p p P Example revsted Applaton of te modfed onvex ull prng to te Example gves sngleton set P ( ) wt te element gven by p ( ) a w/( d ) for x d learly p ( ) p In ts ase generator reeves uplft w [ d /( d )] w s smaller tan uplft mpled by te onvex ull prng and s zero n te lmt as In bot metods te generator beng LNMGU reeves zero proft We note tat ontrary to maret pre obtaned from onvex ull prng te pre p ( ) s dereasng wt load Tat s a trade-off between lower total uplft payment and propertes of te maret pre Example Let s add to te power system desrbed n Example anoter generator wt zero start-up ost: generator g as ost funton C( X) wu ax wt x x and generator g as C( X ) ax wt x x Parameters are assumed to satsfy te followng relatons: a and a are postve onstants wt a w / x a a w / d d x d x Tese ondtons ensure bot tat prmal problem () as unque soluton wt g n a state OFF wt g produng d and tat dual problem (6) results n maret pre set by g w s below a - margnal ost of output by g Clearly g s LNMGU wle g s not Applaton of te onvex ull prng results n sngleton set P wt element p a w / x and uplft of ( a p ) d pad to g Applaton of te modfed onvex ull prng for suffently small results n sngleton set P ( ) wt element p ( ) a w mples zero total uplft payment sne g s not reevng uplft due to p ( ) a w /( d ) We note tat p ( ) p and generators g and g ave zero profts bot n onvex ull prng metod and modfed one In ts example ontrary to onvex ull prng metod w allows nonoperatng generator g to set te pre te new

21 metod results n maret pre set by generator g w as nonzero output n te prmal problem () soluton Tat s owever a spef property of te gven example and n general ase te new metod also allows nonoperatng generator to set te maret pre Example 4 Let s amend Example replang d x by () soluton for outputs and dual problem (6) soluton for a set d x Bot prmal problem P as well as ea generator uplft payment n onvex ull prng metod do not ange However outome of te modfed onvex prng anges and for suffently small s gven by bounded losed nterval P ( ) [ a a w /( d )] We note tat p s below any elements of P ( ) In te lmt as we ave P ( ) [ a a w / d] and agan zero total uplft Tat manfests te fat tat at a pre above a w / d generator g as an opportunty to sgn proftable ontrats wt all te onsumers to supply power volume d and ene f maret pre were set above a w / d ten g would ave to be ompensated for te lost proft Example 5 Let s modfy Example replang d x by d x Sne g as to be ON n entralzed dspat problem and a a we onlude tat g s OFF and g as output equal d n soluton to () Due to a w / x a te onvex ull prng metod produes unque maret pre p a w / x (te set P s sngleton) w mples total uplft w( d / x ) wolly pad to g Relaton a a w / d mples tat for suffently small we ave a a w /( d ) and modfed onvex prng algortm results n unque pre p ( ) a w /( d ) w s ger tan p In te lmt as w / d a) no uplft s pad to g wle g reeves te uplft ( a x It s stragtforward to verfy tat modfed onvex ull prng gves lower total uplft tan onvex ull prng metod VII Conlusons We ave studed unform power prng n one-perod unnode power system wt fxed demand and zero generator mnmal apaty lmts Contrary to onvex ull prng metod w treats ea output volume allowed by generator nternal onstrants as possble even f tat output s tenologally and/or eonomally nfeasble we propose to dentfy - a set of generator output volumes w are bot tenologally and eonomally feasble an be obtaned as a set of solutons to entralzed dspat problem wt generator mal apaty lmts no ger tan tose n te orgnal problem However s not a dret produt of generator orrespondng ndvdual sets Hene utlzaton of would requre ntroduton of new onstrants dependng of more tan one generator output w n turn entals ntroduton of new produts/serves and assoated pres n Lagrange

22 relaxaton proedure To stay wtn sngle ommodty and unform prng framewor nstead of we propose to onsder I w are projetons of nto ndvdual generator nternal feasble sets Tat proedure amounts to ntroduton of a set of new redundant onstrants ea dependng on output of one generator only Te transton from to mples a loss of nformaton sne n general ase annot be reovered just from sets tat means tat ntroduton of new onstrants mxng outputs of dfferent generators and assoated pres may potentally redue total uplft payment even furter Wen alulatng a set of possble output volumes we mposed requrement tat perfetly nelast demand sould be fully ontrated w mples tat power balane onstrant olds for any element of Alternatve approa w was not pursued n te present paper would be to replae fxed demand by onsumer bds wt some beneft funtons eg wt onstant margnal beneft and onsder te lmt as margnal beneft goes to nfnty In ts settng some elements of te set may not belong to te feasble set of te prmal problem Te proposed metod s not just a way to redue te feasble set of te prmal problem to a subset ontanng te optmal ponts Te trval possblty to aeve tat would be to redue generator nternal feasble sets to some small subsets ontanng tose ponts However te resultng set of feasble outputs would not ontan all possble outputs generator may explore stayng wtn tenologal and eonom lmts To ndate n dual problem tat nfntesmal devatons of pre from a gven value may results n under/oversupply of power and ensure nononfsatory prng we enlarge ea set to nlude small negboroods of ea output volume n te ON state of te unt ompatble wt nternal onstrants of te generator and a pont representng an OFF state of te unt Tat algortm results n G ( ) as generator feasble sets n deentralzed dspat problems For te power system under onsderaton te sets and G ( ) an be onstruted expltly Te proposed modfed onvex ull prng approa results n total uplft payment lower tan (or equal) tat n ase of onvex ull prng metod We also sow tat n ase of onvex entralzed dspat problem te proposed prng algortm produes te same pres as onvex ull prng metod - margnal pres Analyss n seton IV entals tat ompared to onvex ull prng metod only LNMGUs e unts wt mal apaty above demand and ost funtons exbtng natural monopoly beavor up to some output volume exeedng demand requre speal treatment n deentralzed dspat problems emergng n dual of te modfed prmal problem and te rest of generatng unts an be onsdered wtout any modfatons e generator deentralzed dspat problems for all oter unts an be formulated as n onvex ull prng metod usng feasble sets defned by generator nternal onstrants Hene f power system as no LNMGU ten proposed metod gves te same set of pres as onvex ull prng proedure Tat observaton owever eavly reles on te assumpton of zero generator mnmal apaty lmts We sowed tat P ( ) a set of maret pres produed by proposed prng metod as well-defned lmt as Also te new approa gves set of maret